Bases of canonical number systems in quartic algebraic number fields

Brunotte, Horst; Huszti, Andrea; Pethő, Attila

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 537-557.

Les systèmes canoniques de numération peuvent être considérés comme des généralisations naturelles de la numération classique des entiers. Dans la présente note, une modification d’un algorithme de B. Kovács et A. Pethő est établie et appliquée au calcul des systèmes canoniques de numération dans certains anneaux d’entiers de corps de nombres algébriques. L’algorithme permet de déterminer tous les systèmes canoniques de numération de quelques corps de nombres de degré quatre.

Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.

[1] S. Akiyama, T. Borbély, H. Brunotte, A. Pethő and J. M. Thuswaldner, On a generalization of the radix representation – a survey, in “High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams”, Fields Institute Commucations, vol. 41 (2004), 19–27. | Zbl

[2] S. Akiyama, T. Borbély, H. Brunotte, A. Pethő and J. M. Thuswaldner, Generalized radix representations and dynamical systems I, Acta Math. Hung., 108 (2005), 207–238. | MR | Zbl

[3] S. Akiyama, H. Brunotte and A. Pethő, Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert, J. Math. Anal. and Appl., 281 (2003), 402–415. | MR | Zbl

[4] S. Akiyama and H. Rao, New criteria for canonical number systems, Acta Arith., 111 (2004), 5–25. | MR | Zbl

[5] S. Akiyama and J. M. Thuswaldner, On the topological structure of fractal tilings generated by quadratic number systems, Comput. Math. Appl. 49 (2005), no. 9-10, 1439–1485. | MR | Zbl

[6] T. Borbély, Általánosított számrendszerek, Master Thesis, University of Debrecen, 2003.

[7] H. Brunotte, On trinomial bases of radix representations of algebraic integers, Acta Sci. Math. (Szeged), 67 (2001), 521–527. | MR | Zbl

[8] H. Brunotte, On cubic CNS polynomials with three real roots, Acta Sci. Math. (Szeged), 70 (2004), 495 – 504. | MR | Zbl

[9] I. Gaál, Diophantine equations and power integral bases, Birkhäuser (Berlin), (2002). | MR | Zbl

[10] W. J. Gilbert, Radix representations of quadratic fields, J. Math. Anal. Appl., 83 (1981), 264–274. | MR | Zbl

[11] E. H. Grossman, Number bases in quadratic fields, Studia Sci. Math. Hungar., 20 (1985), 55–58. | MR | Zbl

[12] V. Grünwald, Intorno all’aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll’aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23 (1885), 203–221, 367.

[13] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné III, Publ. Math. (Debrecen), 23 (1976), 141–165. | MR | Zbl

[14] I. Kátai and B. Kovács, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen, Acta Sci. Math. (Szeged), 42 (1980), 99–107. | MR | Zbl

[15] I. Kátai and B. Kovács, Canonical number systems in imaginary quadratic fields, Acta Math. Acad. Sci. Hungar., 37 (1981), 159–164. | MR | Zbl

[16] I. Kátai and J. Szabó, Canonical number systems for complex integers, Acta Sci. Math. (Szeged), 37 (1975), 255–260. | MR | Zbl

[17] D. E. Knuth, An imaginary number system, Comm. ACM, 3 (1960), 245 – 247. | MR

[18] D. E. Knuth, The Art of Computer Programming, Vol. 2 Semi-numerical Algorithms, Addison Wesley (1998), London 3rd edition. | MR | Zbl

[19] B. Kovács, Canonical number systems in algebraic number fields, Acta Math. Acad. Sci. Hungar., 37 (1981), 405–407. | MR | Zbl

[20] B. Kovács and A. Pethő, Number systems in integral domains, especially in orders of algebraic number fields, Acta Sci. Math. (Szeged), 55 (1991), 287–299. | MR | Zbl

[21] S. Körmendi, Canonical number systems in $\mathbb{Q}\left(\sqrt[3]{2}\right)$, Acta Sci. Math. (Szeged), 50 (1986), 351–357. | MR | Zbl

[22] G. Lettl and A. Pethő, Complete solution of a family of quartic Thue equations, Abh. Math. Sem. Univ. Hamburg 65 (1995), 365–383. | MR | Zbl

[23] M. Mignotte, A. Pethő and R. Roth, Complete solutions of quartic Thue and index form equations, Math. Comp. 65 (1996), 341–354. | MR | Zbl

[24] P. Olajos, Power integral bases in the family of simplest quartic fields, Experiment. Math. 14 (2005), 129–132. | MR | Zbl

[25] A. Pethő, On a polynomial transformation and its application to the construction of a public key cryptosystem, Computational Number Theory, Proc., Walter de Gruyter Publ. Comp. Eds.: A. Pethő, M. Pohst, H. G. Zimmer and H. C. Williams (1991), 31–43. | MR | Zbl

[26] A. Pethő, Notes on CNS polynomials and integral interpolation, More sets, graphs and numbers, 301–315, Bolyai Soc. Math. Stud., 15, Springer, Berlin, 2006. | MR | Zbl

[27] A. Pethő, Connections between power integral bases and radix representations in algebraic number fields, Proc. of the 2003 Nagoya Conf. “Yokoi-Chowla Conjecture and Related Problems”, Furukawa Total Pr. Co. (2004), 115–125.

[28] R. Robertson, Power bases for cyclotomic integer rings, J. Number Theory, 69 (1998), 98–118. | MR | Zbl

[29] R. Robertson, Power bases for 2-power cyclotomic integer rings, J. Number Theory, 88 (2001), 196–209. | MR | Zbl

[30] K. Scheicher, Kanonische Ziffernsysteme und Automaten, Grazer Math. Ber., 333 (1997), 1–17. | MR | Zbl

[31] D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137–1152. | MR | Zbl

[32] J. M. Thuswaldner, Elementary properties of canonical number systems in quadratic fields, in: Applications of Fibonacci Numbers, Volume 7, G. E. Bergum et al. (eds.), Kluwer Academic Publishers, Dordrecht (1998), 405–414. | MR | Zbl